LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process

2025/07/30 by Feng, Xiaodong, Guo, Ling, Wan, Xiaoliang +3 · 1 citation
#Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML)

paper · doi:10.48550/arxiv.2507.22493

Abstract

We propose a novel probabilistic framework, termed LVM-GP, for uncertainty quantification in solving forward and inverse partial differential equations (PDEs) with noisy data. The core idea is to construct a stochastic mapping from the input to a high-dimensional latent representation, enabling uncertainty-aware prediction of the solution. Specifically, the architecture consists of a confidence-aware encoder and a probabilistic decoder. The encoder implements a high-dimensional latent variable model based on a Gaussian process (LVM-GP), where the latent representation is constructed by interpolating between a learnable deterministic feature and a Gaussian process prior, with the interpolation strength adaptively controlled by a confidence function learned from data. The decoder defines a conditional Gaussian distribution over the solution field, where the mean is predicted by a neural operator applied to the latent representation, allowing the model to learn flexible function-to-function mapping. Moreover, physical laws are enforced as soft constraints in the loss function to ensure consistency with the underlying PDE structure. Compared to existing approaches such as Bayesian physics-informed neural networks (B-PINNs) and deep ensembles, the proposed framework can efficiently capture functional dependencies via merging a latent Gaussian process and neural operator, resulting in competitive predictive accuracy and robust uncertainty quantification. Numerical experiments demonstrate the effectiveness and reliability of the method.

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Cited by

• MAD-NG: Meta-Auto-Decoder Neural Galerkin Method for Solving Parametric Partial Differential Equations

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